The mean spherical model for Ising spin systems of Lewis and Wannier replaces the condition that each spin variable ${\sigma }_i=\pm \frac{1}{2}$ by the weaker condition that $\Sigma <{{\sigma }_i}^2>=\frac{1}{4}\Omega$, where $\Omega =\mathrm{number}\mathrm{of}\mathrm{lattice}\mathrm{sites}$. This model has the same properties, in the thermodynamic $\mathrm{limit}\Omega \to \infty$, as the spherical model of Berlin and Kac, and is immediately applicable, by a well-known isomorphism, to lattice gases with an interparticle potential $v\left(\mathbf{r}\right)$ of the form $v\left(\mathbf{r}\right)=\infty$ for r=0 (no multiple occupancy of the same lattice site), $v\left(\mathbf{r}\right)$ finite for $\mathbf{r}\ne 0$. We have now extended this model to more general lattice gases where $v\left(\mathbf{r}\right)=\infty$ for r in some domain $D$, i.e., lattice gases of particles with extended hard cores. This permits extension of the model to continuum systems. We find, for this model, that the direct correlation function of Ornstein and Zernike is equal to $-\beta v\left(\mathbf{r}\right) (\beta$ the reciprocal temperature) for r not in $D$, and is determined for r in $D$ by the requirement that the two-particle distribution functions $n_2({\mathbf{r}}_1,{\mathbf{r}}_2)$ vanish for ${\mathrm{r}}_{1,2}$ in $D$. All higher order (modified) Ursell functions (spin semi-invariants) vanish for the model. The model thus yields the same pair distribution function as the Percus-Yevick integral equation for the case when $v\left(\mathbf{r}\right)=0\mathrm{}$ for r not in $D$, giving also, incidentally, an upper bound to the density for which solutions of this equation exist. The thermodynamic properties of this model are also discussed and it is shown that the partition function becomes singular in the continuum limit.

• Received 18 October 1965

DOI:https://doi.org/10.1103/PhysRev.144.251